ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-11

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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A $3 \times 3$ inverse computed by row reducing $[A ∣ I]$

Example

For $A = (\begin{array}{ccc} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{array}),$ row reduction of $[A ∣ I_{3}]$ gives $A^{- 1} = (\begin{array}{ccc} 1 & - 1 & 1 \\ 0 & 1 & - 1 \\ 0 & 0 & 1 \end{array}) .$

Facts & Assumptions

Given: The displayed real matrix $A$ .

[L1]

Reducing $[A ∣ I]$ to $[I ∣ B]$ gives $B = A^{- 1}$ (Row reducing $[A ∣ I]$ yields $[I ∣ A^{- 1}]$ exactly when $A$ is invertible).

[L2]

Matrix multiplication is associative and unital (Matrix multiplication is associative, unital, distributive, and compatible with scalar multiplication).

Verification

technique · direct

1.1

Starting from $[A ∣ I_{3}]$ , replace row $1$ by row $1$ minus row $2$ , then replace row $0$ by row $0$ minus the new row $1$ . The left block becomes $I_{3}$ and the right block becomes the displayed matrix $B$ .

L1algebra

2.1

Direct multiplication gives $A B = I_{3}$ and $B A = I_{3}$ : the only off-diagonal sums are $- 1 + 1$ , $1 - 1$ , and $- 1 + 1$ . Hence $B = A^{- 1}$ .

step 1.1L2algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

D. Margalit and J. Rabinoff, Interactive Linear Algebra, §3.6 (standard reference, not scraped)

A 3×33×3 inverse computed by row reducing [A∣I][A∣I]